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In this paper we investigate classical solution of a semi-linear system ofbackward stochastic integral partial differential equations driven by aBrownian motion and a Poisson point process. By proving an It\^{o}-Wentzellformula for jump diffusions as well as an abstract result of stochasticevolution equations, we obtain the stochastic integral partial differentialequation for the inverse of the stochastic flow generated by a stochasticdifferential equation driven by a Brownian motion and a Poisson point process.By composing the random field generated by the solution of a backwardstochastic differential equation with the inverse of the stochastic flow, weconstruct the classical solution of the system of backward stochastic integralpartial differential equations. As a result, we establish a stochasticFeynman-Kac formula.

Semi-linear Backward Stochastic Integral Partial Differential Equations driven by a Brownian motion and a Poisson point process